A letter concerning Leonetti's paper `Continuous Projections onto Ideal Convergent Sequences'

TL;DR

This paper offers a shorter proof of a general result showing that certain quotient spaces of bounded sequences do not embed into the space of all bounded sequences, extending Leonetti's earlier work on ideal convergence.

Contribution

It provides a more concise proof and a broader generalization of Leonetti's result on the non-embeddability of specific quotient spaces related to ideal convergence.

Findings

The quotient space _ / c_{0,} does not embed into .

The proof simplifies and generalizes Leonetti's original argument.

The result applies to a wider class of ideals with uncountable families.

Abstract

Leonetti proved that whenever $\mathcal I$ is an ideal on $\mathbb N$ such that there exists an~uncountable family of sets that are not in $\mathcal I$ with the property that the intersection of any two distinct members of that family is in $\mathcal I$, then the space $c_{0,\mathcal I}$ of sequences in $\ell_\infty$ that converge to 0 along $\mathcal I$ is not complemented. We provide a shorter proof of a more general fact that the quotient space $\ell_\infty / c_{0,\mathcal I}$ does not even embed into $\ell_\infty$.